All graphs in this paper are finite and simple. For general graph theory notation we follow [1]}. A \(q\) -coloring of a graph \(G\)  assigns a color from \(\lbrace 1,\ldots ,q\rbrace \)  to each vertex of \(G\)  such that adjacent vertices are assigned different colors. We say that a graph \(G\)  is \(q\) -colorable if \(G\)  admits a \(q\) -coloring.
The chromatic number of a graph \(G\) , denoted by \(\chi (G)\) , is the minimum number \(q\)  for which \(G\)  is \(q\) -colorable.
The clique number of \(G\) , denoted by \(\omega (G)\) , is the size of a largest clique in \(G\) .
A graph \(G\)  is perfect if \(\chi (H)=\omega (H)\)  for each induced subgraph \(H\)  of \(G\)
